1. Rendering General BSDFs and BSSDFs Randy Rauwendaal

2. BSDFs Bidirectional Scattering Distribution Function Gives a mathematical description of the way that light is scattered by a surface Is a generalization of a BRDF and a BTDF

3. The Bidirectional Scattering Distribution Function The BSDF is a mathematical description of the light-scattering properties of a surface Let L o (ω o ) denote the radiance leaving x in direction ω o The light strikes the surfaces and generates an irradiance It can be observed experimentally that as dE(ω i ) is increased there is a proportional increase in the observed radiance dL o (ω o ) The BSDF is now defined to be this constant of proportionality

4. The Scattering Equation By integrating the relationship Over all directions, we can now predict L o (ω o ) This is summarized by the (surface) scattering equation, This equation can be used to predict the appearance of the surface, given a description of the incident illumination

5. The BRDF and BTDF Usually scattered light is subdivided reflected and transmitted components, which are treated separately –Bidirectional Reflectance Distribution Function (BRDF), denoted f r –Bidirectional Transmittance Distribution Function (BTDF), denoted f t The BRDF is obtained by simply restricting f s to the domain The BTDF is similarly obtained by restricting f s to the domain Thus the BSDF is the union of two BRDF’s (one for each side of the surface), and two BTDF’s (one for light transmitted in each direction). –More convenient, 1 function instead of 4 –Purely reflective or transmissive surfaces are special case of this formulation

6. Properties of the BRDF BRDF’s that describe real surfaces have basic properties –Symmetry –Energy Conservation These properties are unique to reflection –It cannot be assumed that other distribution functions satisfy these properties

7. The BSDF for Specular Reflection For a perfect mirror, the desired relationship between L i and L o is that Where M N (ω o ) is the mirror direction, obtained by reflecting around the normal N Now we define this BSDF in terms of a special Dirac distribution δ σ┴, we is defined by the property that The Dirac distribution (or delta function) also has the properties that –δ(x) = 0 for all x ≠ 0 –∫ R δ(x)dx = 1 Which implies the useful identity

8. The BSDF for Specular Reflection Now we can write the equation for a BSDF for a perfect mirror By expanding the definition of the projected solid angle, we can write the scattering distribution function in the form Which allows us to write the mirror BSDF as Expressions containing Dirac distributions must be evaluated with great care, particularly when the measure functions associated with the Dirac distribution is different than the measure function used for integration

9. The BSDF for Refraction We define a mapping Such that R(ω i ) is the transmitted direction corresponding to the incident direction ω i Given this mapping, the relation between Li and Lt due to refraction can be expressed as The corresponding BSDF is thus This functions expresses the relationship between ω i and ω t, and also the fact that the radiance is scale by a factor of (η t /η i ) 2

10. Reciprocity and Conservation Laws for General BSDF’s The BSDF of any physically plausible material must satisfy Where η i and η o are the refractive indices of the materials containing ω i and ω o respectively This is a generalization of the BRDF property of symmetry We also investigate how light scattering is constrained by the law of conservation of energy, and we derive a simple condition that must be satisfied by any BSDF that is energy conserving

11. A Reciprocity Principle for General BSDF’s To prove a reciprocity condition for general BSDF’s, we consider the light energy scattered between to directions ω i and ω o at a point x in an isothermal enclosure By the principle of detailed balance, the rates of scattering from ω i to ω o and from ω o to ω i are equal (dΦ 1 = dΦ 2 ), while by Kirchoff’s equilibrium radiance law, the incident radiance from each direction is proportional to the refractive index squared Putting these facts together we get the desired reciprocity condition

12. Conservation of Energy Theorem If f s is the BSDF for a physically valid surface, which is either the boundary of an opaque object or the interface between two non-absorbing media, then Proof Where E denotes the irradiance, and M denotes the radiant exitance Considering the radiance distribution, we let the incident power be concentrated in a single direction ω i From which the requirement E ≤ M gives the desired result

13. BSSDFs Bidirectional Surface Scattering Distribution Function There not a whole lot material on general BSSDFs, so instead we’ll focus on…

14. BSSRDFs Bidirectional Surface Scattering Reflectance Distribution Function The BSSRDF relates the outgoing radiance to the incident flux The BRDF is an approximation of the BSSRDF for which it is assumes that light enters and leaves at the same point The outgoing radiance is computed by integrating the incident radiance over incoming directions and area, A

15. Symbol Reference

16. The Diffusion Approximation The diffusion approximation is based on the observation that the light distribution in highly scattering media tends to become isotropic The volumetric source distribution can be approximated using the dipole method The dipole method consists of positioning two point sources near the surface in such a way as to satisfy the required boundary condition The diffuse reflectance due to the dipole source can be computed (with much hand waving) as Taking into account the Fresnel reflection at the boundary for both the incoming light and the outgoing radiance Where S d is the diffusion term of the BSSRDF, which represents multiple scattering

17. The Diffusion Approximation An incoming ray is transformed into a dipole source for the diffusion approximation

18. Single Scattering Term The total outgoing radiance, due to single scattering is computed by integrating the incident radiance along the refracted outgoing ray The single scattering BSSRDF is defined implicitly by the second line of this equation Single scattering occurs only when the refracted incoming and outgoing rays intersect, and is computed as an integral over path length s along the refracted outgoing ray

19. The BSSRDF Model The complete BSSRDF model is a sum of the diffusion approximation and the single scattering term This model accounts for light transport between different locations on the surface, and it simulates both the directional component (due to single scattering) as well as the diffuse component (due to multiple scattering)

20. BRDF Approximation We can approximate the BSSRDF with a BRDF by assuming that the incident illumination is uniform By integrating the diffusion term we find the total diffuse reflectance of the material The integration of the single scattering term for a semi-infinite medium gives The complete BRDF model is the sum of the diffuse reflectance scaled by the Fresnel term and the single scattering approximation This model has the same parameters as the BSSRDF Note that the amount of light is computed from the intrinsic material parameters The BRDF approximation is useful for opaque materials, which have a very short mean free path

21. Rendering Using the BSSRDF The BSSRDF model derived only applies to semi-infinite homogeneous media, for a practical model we must consider –Efficient integration of the BSSRDF (importance sampling) –Single scattering evaluation for arbitrary geometry –Diffusion approximation for arbitrary geometry –Texture (spatial variation on the object surface)

22. BRDF vs BSSRDF

25. References Eric Veach, Robust monte carlo methods for light transport simulation, 1997 Jensen, H. W., Marschner, S. R., Levoy, M., and Hanrahan, P. 2001 A practical model for subsurface light transport. In Proceeding of SIGGRAPH 2001, 511-518 Henrik Wann Jensen and Juan Buhler. A rapid hierarchical rendering technique for translucent materials. ACM Transactions on Graphics, 21(3):576.581, July 2002.